We touched upon some aspects relating to the combinatorics of list decoding Reed-Solomon codes. We discussed the geometric intuition underlying Reed-Solomon decoding, and presented Sudan’s algorithm for polynomial reconstruction/list decoding RS codes. We analyzed it by optimizing individual degrees leading to an algorithm working for agreement parameter where is the degree and is the number of points. By optimizing the -weighted degree, we improved this to .

We motivated the method of using multiple zeroes in the interpolation with an example, and then presented the details of how to enforce multiple (say ) zeroes at a point for a bivariate polynomial, why it buys us zeroes for each point of agreement, and why this should improve the agreement needed by a further factor thus matching the Johnson bound.

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